inﬂuenceoflongitudinalcreepageandwheelinertiaon short ...jbarber/jet.pdf172 mciavarellaandjbarber...

SPECIAL ISSUE PAPER 171

Influence of longitudinal creepage and wheel inertia onshort-pitch corrugation: a resonance-free mechanism toexplain the roaring rail phenomenonM Ciavarella1∗ and J Barber2

1CEMEC-DIASS, Politecnico di Bari, Bari Italy2University of Michigan, MI, USA

The manuscript was received on 28 September 2007 and was accepted after revision for publication on 11 January 2008.

DOI: 10.1243/13506501JET373

Abstract: Short-pitch corrugation (30–80 mm in wavelength) in railways, despite being wellknown since the early days of the railways because of its criticality in producing damage, ‘roaringrail’ or ‘howling wheel’ noise, and indirectly rolling contact fatigue, is considered an enigmaticphenomenon. In fact, most available data seem to show a non-linearly increasing wavelengthwith speed, and an almost fixed wavelength, while most models based on system resonancespredict a fixed frequency. More enigmatic still, many data points fall in a range of frequencieswhere there is no evident resonance in the wheel–railtrack system (the large gap between thelow frequencies resonances from 50 to 300 Hz and the very high pinned–pinned mode reso-nant frequencies which correspond generally to 850–1100 Hz in railways. Yet the most commonclassifications of corrugation continue to associate corrugation to frequency-fixing mechanisms.

Johnson’s early studies on the Hertz normal spring resonance suggest that plasticity-basedrepeats impact mechanism, or differential wear mechanism both seemed to be not appropriateto explain short-pitch corrugation. In particular, longitudinal creepage (obviously associatedwith braking or acceleration very common on uphill grades, near stations, but also in curveswhere profiles provide insufficient steering capability) seemed to act to suppress corrugation,rather than promoting it, as suggested in the model of Grassie and Johnson. Only a few, verycomprehensive models that include all the relevant receptances consider the effect of wheelinertia: indeed, these models indicate many possible corrugation regimes and, in particular,point at lateral creepage mechanisms at the pinned–pinned resonant frequency as giving muchlarger growth than longitudinal creepage, so the possibility of a corrugation regime independentof wheelset or railtrack resonances has largely remained hidden, despite it being present in someresults.

In this paper, a simple model that returns to a pure longitudinal creepage mechanism issuggested, showing that it is essential to include the rotational dynamics of the wheel in thesystem, similar to Grassie and Johnson’s model. In particular, a simple full-stick Winkler-contactmechanics model can estimate the effect of transient contact mechanics. For typical inertias,the conditions are closer to the constant tangential load (which is the correct limit at zero speedanyway) and seem to explain the basic features of wear-induced instability in the existing exper-imental data. For larger inertias, which may also be possible for heavy wheelsets, the modelpredicts results closer to Grassie and Johnson’s assumption of constant creepage, i.e. only a lim-ited range of possible short-pitch corrugation. The model also suggests that although the growthof corrugation depends strongly on the amplification of the normal load, the wavelength of thismode of corrugation depends very little on the vertical resonances of the systems, so that it would

∗Corresponding author: CEMEC-DIASS, Politecnico di Bari,

Vle Gentile 182, Bari 70126, Italy. email: [email protected]

JET373 © IMechE 2008 Proc. IMechE Vol. 222 Part J: J. Engineering Tribology

172 M Ciavarella and J Barber

persist even in a model with no resonance altogether. It is possible that the exact frequency of thisregime depends on the details of the contact geometry, here simplified using the Winkler model.

Finally, a reason why this mechanism of longitudinal creepage corrugation, despite perhapsgiving 10–20 times apparently lower growth than lateral creepage, may indeed be the correctmechanism to interpret the classical data, is that longitudinal creepage can be 10 times higherthan lateral (5 per cent instead of 0.5 per cent), and as corrugation growth is proportional tosquare of creepage, there is a factor 100 that largely compensates for this. There is still someprogress to be made to obtain a reliable model to compare the various regimes, but clearly thisregime should be considered when devising remedies to corrugation.

Keywords: corrugation, transient non-linear dynamic, perturbation, railways

1 INTRODUCTION

The application motivating this investigation is thestudy of ‘corrugations’ on the running surface causedby the action of the railways wheels, a phenomenonthat has been observed throughout railway history,but not fully understood [1], particularly for short-pitch rail corrugation (‘roaring rails’) in the range of20–80 mm wavelength, which seems to show a con-stant wavelength or a non-proportional increase ofcorrugation wavelength with increasing train speed.This common belief comes from the large amountsof data available in BR reports of 1911, in DavidHarrison’s thesis [2], a figure from which is reproducedin the well-known paper by Grassie and Kalousek [1],and in the more recent paper by Bhaskar et al. [3],adapted here in Fig. 1.

In Harrison’s data are particularly surprisingbecause they seem to suggest an almost ‘fixed wave-length’, whereas the other results are linear in the givenrange, and do not start from the origin, suggestingperhaps some non-linear effect, or a threshold.

An important study was that of Grassie and Johnson[4], which despite using a simplified two-dimensionalformulation and a ‘Winkler’ model for the contactproblem, contained quite a few features and hencewould be expected to give qualitatively reasonable pre-dictions. However, unfortunately, one assumption wasmade that the authors of the current paper believe wascritically incorrect: that longitudinal creepage wouldbe constant, perhaps based on the more reasonableassumption that the large mass of the vehicle wouldmake its speed constant. Grassie and Johnson foundno mechanism for corrugation in the range of interestof Fig. 1 and actually suggested that this could be amechanism for suppression of corrugation.

In many later studies, the models introduced manydetailed improvements, although those modifying theassumption of constant longitudinal creepage gener-ally include so many other effects that it is difficultto distinguish the relative roles of each, and theinvestigations are generally relative only to a specialsystem under a certain range of conditions, so that the

simplicity of the original Grassie and Johnson modelto identify the main relevant features was lost.

For example, Frederick [5] introduced an elegantperturbation analysis via complex transfer functions,which makes it simple to check if the phase of themaximum dissipation would produce amplificationof the original corrugation in the frequency rangeof interest. He studies both lateral and longitudinalcreepage assuming for the wheel receptance the sameconcentrated mass model, similar to what is donein this paper, so the main difference between lat-eral and longitudinal creepage is the receptance ofthe rail (a pure damper longitudinally and a quiteflexible structure laterally). The results show a dom-inant lateral creepage mechanism at pinned–pinnedresonance, but also other possible regimes, includ-ing one with longitudinal creepage not correspondingto a resonance (at about 750 Hz) and only activeabove sleepers. It must be said that Frederick’s modelassumes a wear parameter that is not energy dissi-pation but more simply the Clayton creep-force one,and that it shows results only for a quite high veloc-ity (40 or 60 m/s) and restricted to frequencies above300 Hz, whereas here the entire range of velocitiesand frequencies is investigated. Also, the assump-tions of saturated creep changes the relative effectsof fluctuation of creepage and tangential force withrespect to the more general condition at small creep-age in the linear regime. A similar assumption is madehere.

It must be also said that while many authors con-sider the wheelset resonances very narrow, perhapsremoving them from the analysis, they are less cau-tious when dealing with the pinned–pinned reso-nance, which is also quite delicate to deal with – anerror in the damping factor may easily change theamplitude response, and also the different patternof corrugation along the sleeper bay is not alwaysobserved. Also, despite that the presence of the peri-odic structure does seem to trigger the corrugationmechanism [5], the pinned–pinned resonance fre-quency is not as frequently observed as the data inFig. 1 suggest.

Proc. IMechE Vol. 222 Part J: J. Engineering Tribology JET373 © IMechE 2008

Influence of longitudinal creepage and wheel inertia 173

Fig. 1 Experimentally observed variation of corruga-tion wavelength with speed – adapted fromBhaskar et al [3]. Superposed are the constant fre-quencies of 300, 1100, and 1700 Hz, which someauthors indicate as the resonances of the systemwhich should justify corrugation

Tassilly and Vincents [6, 7] further developedFrederick’s model, applying it to the Paris metrosystem, and also adding the low-frequency resonancesincluding the wheelset ones experimentally measured.In this case, there are few possibilities to attributecorrugation to one of these resonance frequencies,but ultimately they found predominantly transver-sal wear on the leading wheelset corresponding toits first bending mode (a lateral mechanism), andlongitudinal wear on the rear one, related to the firsttorsional mode of the wheelset. Tassilly and Vincent’s‘case 3, corrugation on a concrete track’, however, doesshow a possible regime of corrugation independentof resonances. Elkins et al. [8] found that the 300 Hz(second) torsional resonance for the wheelset may beresponsible for some types of corrugation.

The effect of discrete sleepers is also included byHempelmann [9] (see also reference [10]), who usea linearization over the full non-linear model forcreepage law, and return to the energy dissipationparameter. They also include sophisticated numericaltreatments developed to deal with transient contacteffects by Gross–Thebing [11]. Unfortunately, theyneglect longitudinal creepage, so have receptancesonly in the lateral direction. Similarly to Frederick,they find corrugation growth in almost the entirerange from 0 to 2000 Hz (except at the pinned–pinnedresonance) at midspan, whereas above sleepers theyfound a regime around the 350 Hz (which was500–600 Hz in Frederick’s study) and more impor-tantly at the pinned–pinned resonance frequenciesand higher where the magnitude is much higherthan the other regime. Only the presence of thepinned–pinned regime has generally been considered

as Hempelmann’s results∗, and not the underlyingmechanism for growth clearly not associated to res-onances of the system – which may perhaps be theonly left in a system with continuously laid track, orwhen longitudinal creepage is considered instead oflateral, especially as longitudinal creepage is gener-ally much larger than the lateral (see in particularthe illuminating discussion which is a part of theHempelmann and Knothe 1996 paper by Frederick).Moreover, even if is assumed that growth initiates atthe sleepers, probably the process continues througha continuous range of wavelengths depending on posi-tion. The large collection of data in Fig. 1 could not beexplained by the pinned–pinned range which for prac-tical speeds is concentrated in the 20–30 mm range,but could also be the result of some averaging. How-ever, it is very likely that longitudinal creepage playsa role; for example, Vancouver SkyTrain’s short-pitchcorrugation was largely unaffected by sleeper spacing[3]. So why could Grassie and Johnson’s model not findsuch a mechanism?

In principle, Muller [12, 13] extends theHempelmann model. However, he omits the dynamicmotion of wheel and tends to find agreement withHempelmann’s results, suggesting mainly pinned–pinned resonance corrugation, with perhaps othersecondary regimes at 300 and 1700 Hz, as indicatedin Fig. 1 with the three lines clearly not sufficient toexplain the experimental values.

The conclusion of Grassie and Johnson [4] thatlongitudinal creepage would more likely suppress cor-rugation than promote it in the short-pitch rangewas never really reassessed, as Hempelmann hadneglected longitudinal creepage, and although someauthors do include wheelset receptances (Frederick,Tassilly and Vincents, etc.) and seem to indicate lon-gitudinal creepage mechanisms, it is a little unclear ifthese are necessarily to be associated to resonances ofthe system or not.

Bhaskar et al. [3] also looked at the problem, butfrom a different perspective of spin-creepage corruga-tions as a result of closely conformal wheel and railprofiles. They quite honestly do not seem happy withthe prediction of their model, which however neglectswheelset receptances (and hence all inertia effects)and otherwise could have been compared with that ofHempelmann [9], and Hempelmann and Knothe [10].

In the present paper, returning to the simple caseof longitudinal-creepage and reconsidering the anal-ysis of Grassie and Johnson [4], the assumption of

∗A second regime is suggested by Hempelmann and Knothe between

200 and 450 Hz regime for systems with high pad-stiffness. This

must be the regime considered in Hempelmann’s original thesis [9]

whose chapter 7 rather points at corrugation being highest in the

regimes of 400 and 1450 Hz.



constant longitudinal creepage is substituted by, therotational inertia of the wheel. It is demonstrated thatthis simple modification is sufficient to make predic-tions that are quite close to experimental observations.The attempts of more sophisticated models shouldtherefore perhaps introduce longitudinal creepage,unless there is clear evidence of its absence, and con-sequently of wheel inertia and rotational dynamics.

2 THE MODEL

A simple model very close to that of Grassie andJohnson [4] is attempted here, but including wheelinertia and using a linear perturbation approach.The Winkler or wire-brush model assumes thatthe tangential tractions are proportional to thedisplacements

q = kqux (1)

where kq is a Winkler modulus which must be chosenby some best-fit criterion. This leads to a consider-able simplification, since all coupling effects leadingto integral equations are eliminated. All other aspectsof the model are simplified as much as possible,since the paper seeks merely to demonstrate that theeffect of wheel inertia is critical to the generationof a regime of corrugation – which has so far hasescaped attention – and which here is found only withqualitatively realistic predictions. In particular, atten-tion is restricted to two dimensions and the modelused, taken from the literature and representativeonly of an intercity track, describes the rail dynam-ics. Clearly these approximations are very restrictiveand exclude many effects, such as the pinned–pinnedresonance due to sleeper spacing, interaction of adj-acent wheelsets, three-dimensional (lateral) effects: allthese effects may be more able to generate alterna-tive regimes of corrugations. A quantitatively accuratemodel which could assess the relative importance ofall these modes has so far not been produced. Hence,the present model is introduced to produce evidenceof the regime of corrugation which depends on onlya few geometrical and one loading parameter (meannormal load), making the qualitative effect of theseparameters easy to assess.

It is assumed that the vehicle is moving to theleft at constant speed V , so that the wheel is rotat-ing counter-clockwise at some speed �. A rigid-bodyvelocity V is superposed to the right, bringing the cen-tre of the wheel to rest, and causing the rail to moveat speed V to the right. Assuming that the vehicle isbraking, the friction force on the wheel opposes thedirection of motion V and hence is to the right, andthe braking torque B opposes the direction of rota-tion, as shown in Fig. 2. This figure also shows the

Fig. 2 The wheel-rail contact point and coordinatesystem

sign convention for the coordinate x and for the elasticdisplacement ux of a point on the wheel in the contactzone.

If there was no elastic deformation, the velocity ofa point on the wheel in the contact region in thex-direction would be �R. However, the steady-statetensile strain ∂ux/∂x and the elastic displacement vari-ation in time add so that the rightward velocity of apoint on the wheel is

vx =(

1 + ∂ux

∂x

)�R + ∂ux

∂t(2)

All the elastic deformation in concentrated in thewheel, which means that an equivalent modulus forthe wheel is used treating the rail (or the surface) asrigid. In that case, points on the rail move at constantspeed V and a slip velocity

sx = V − �R − V∂ux

∂x− ∂ux

∂t(3)

can be defined in the direction that points on the rail,slipping to the right relative to the wheel. Further, sup-pose that the coefficient of friction is sufficiently highto prevent slip occuring anywhere: sx = 0 and hence

V∂ux

∂x+ ∂ux

∂t= V − �(t)R (4)

The wheel may not spin at a constant rate due to rota-tional dynamics and it is assumed that �(t) is a knownfunction and, in fact, it will typically have the form

�(t) = �0 + �1 exp(ıωt) (5)

The general solution of equation (4) is

ux(x, t) = f (x − Vt) + Vt − �0Rt + ı�1Rω

exp(ıωt)

(6)

where f is any arbitrary function of (x − Vt). This func-tion is determined in practice by the conditions at the



leading edge of the contact, where the instantaneousstate of the surface is ‘locked-in’ to a state of stick.

For the boundary condition, it is assumed thatthe shear traction at the leading edge is zero, i.e.qx(−a) = 0, and hence in view of the Winkler founda-tion

ux(−a, t) = 0 (7)

at all times t .Notice that in a full three-dimensional elasticity

solution, there will be tensile strains ahead of the con-tact and hence the elastic displacement at the leadingedge will be non-zero. However, the Winkler assump-tion is the basis of the ‘simplified’ theory of Kalker [14]and also of his numerical algorithm‘FASTSIM’ [15] andhas been shown to agree quite well with the more exacttheory in other respects.

In the steady state, there is no dependence on t and�1 = 0. Then

f (−a − Vt) + Vt − �0Rt = 0 (8)

Writing η = −(a + Vt); t = −(a + η)/V , then f (η) =(V − �0R)(a + η)/V , and for the displacement ux , thelinearly varying term from zero is recovered at theleading edge of the Carter [16] classical solution

ux = (V − �0R)(a + x − Vt)V

+ Vt − �0Rt

= ξ0(a + x) (9)

see Barber [17], where a steady-state creepage ratio isintroduced

ξ0 =(

1 − �0RV

)(10)

Since the equations are linear, the oscillatory solu-tion can be written as the sum of the steady-statesolution and an oscillatory term which has to satisfythe equation

uωx = f ω(x − Vt) + ı�1R

ωexp(ıωt) (11)

with boundary condition uωx (−a, t) = 0 at all times t .

Applying the boundary condition, the complete tran-sient solution is then

ux = ξ0(a + x) + ı�1Rω

exp(ıωt)

×[

exp(

− ıω(a + x)

V

)− 1

](12)

It is easily verified that this expression satisfies theoriginal governing equation (4) and the boundary con-dition (7) at all times t . The displacement must of

course be a real quantity, and so the real part of this hasto be extracted to describe the physics of the problem.

The time-varying tangential force Q(t) is given by

Q(t) =∫ a

−aqx(x)dx = kq

∫ a

−aux(x)dx (13)

from equation (1) and substituting for ux fromequation (12) and performing the integration, Q(t) =Q0 + Q1 exp(ıωt), where

Q0 = 2kqξ0a2 ; Q1 = 2kq�1Ra

ω

{V

2ωa

×[

1 − exp(

−2ıωaV

)]− ı

}(14)

2.1 Effect of varying normal force

So far, only oscillations in tangential load and angularvelocity have been considered, but the driving termof the present problem is an oscillation of the normalforce in time, due to some existing corrugation

� exp(ıωt)

where ω = 2πf = 2πV /λ and λ is the wavelength ofthe corrugation. In fact, the objective of the analysis isto determine which if any wavelengths lead to oscilla-tions such that the resulting wear occurs at the troughsof the corrugation and hence amplification of an initialperturbation.

The corresponding oscillation in the normal contactforce P(t) will cause the semi-length of the contact ato oscillate, so

a(t) = a0 + a1 exp(ıωt) (15)

Equation (6) remains unchanged when a is non-constant, but the boundary condition (7) becomes

ux(−a(t), t) = 0 (16)

giving

f (−a0 − a1 exp(ıωt) − Vt) + Vt

− �0Rt + ı�1Rω

exp(ıωt) = 0 (17)

Since a1 � a0, the first term is expanded as

f (−a0 − a1 exp(ıωt) − Vt)

= f (−a0 − Vt) − f ′(−a0 − Vt)a1 exp(ıωt) (18)

and the complete function as the form f (η) = f0(η) +f1(η), where f1 � f0, and f0(η) = (1 − �0R/V )(a0 + η),



and f ′ can be replaced by f′

0 = (1 − �0R/V ). Then,writing η = −(a0 + Vt); t = −(a0 + η)/V

f (η) − a1f ′(η) exp(ıωt)

−(

1 − �0RV

)(a0 + η) + ı�1R

ωexp(ıωt) = 0 (19)

and hence

f1(η) = a1

(1 − �0R

V

)exp

(− ıω(a0 + η)

V

)

− ı�1Rω

exp(

− ıω(a0 + η)

V

)(20)

The displacement ux is then recovered from equation(6) as

ux =(

1 − �0RV

)(a0 + x) + ı�1R

ω

×[

1 − exp(

− ıω(a0 + x)

V

)]exp(ıωt)

+ a1

(1 − �0R

V

)exp

(− ıω(a0 + x)

V

)exp(ıωt)

(21)

2.2 Tangential load and dissipation

To obtain the tangential load for the general case justsolved of varying angular speed and normal load, thefollowing has to be calculated

∫ a

−aux(x)dx ≈

∫ (a0+a1 exp(ıωt)

−(a0+a1 exp(ıωt)

(1 − �0R

V

)(a0 + x) dx

+ exp(ıωt)∫ a0

−a0

[ı�1R

ω

[1 − exp

(− ıω(a0 + x)

V

)]

+ a1

(1 − �0R

V

)exp

(− ıω(a0 + x)

V

)]dx

where only first-order terms in the perturbation havebeen included. The first integral is trivial, and the terminvolving �1 is identical with that for constant a andcontributes a term

2�1Ra0

ω

{V

2ωa0

[1 − exp

(−2ıωa0

V

)]− ı

}exp(ıωt)

from equation (14). Finally, for the term involving a1

∫ a0

−a0

exp(− ıω(a0 + x)

V

)dx = − V

ıω

[exp

(−2ıωa0

V

)− 1

](22)

The dissipation rate under full stick conditions, isgiven by the release of the energy at the trailing edge,

which can be found from

D(t) = 12

Vkq[ux(a(t))]2 = 12

Vkq[ux(a0 + a1 exp(ιωt))]2

where, substituting equation (15) into equation (21)and dropping second-order terms gives

ux(a(t)) = 2ξ0a0 + ξ0a1 exp(ιωt)

+[ι�1R

ω+

(ξ0a1 − ι�1R

ω

)exp

(−2ιωa0

V

×(

1 + a1

2a0exp ιωt

))]exp (ιωt) (23)

The dissipation rate can therefore be written asD(t) = D0 + D1 exp(ıωt), and collecting results for Qand D

Q0 = 2kqξ0a20 D0 = V ξ0Q0 (24)

and

Q1 = 2ξ0kqa0a1

{1 − ı

ζ

(1 − exp(−ıζ )

)}

− �14a2

0Rkq

V1ζ

{1ζ

(1 − exp(−ıζ )

) − ı

}(25)

D1 = 4kqV ξ 20 a0a1

(1 + exp (−ıζ )

)2

+ 4kqV ξ0a20

2i�1RV

(1 − exp(−ıζ )

)2ζ

(26)

where the parameter is defined as

ζ = 2ωa0

V= 4πa0

λ(27)

which represents the extent of the initial corrugation(in radians) present in the contact area at any giventime.

The term a1 can be related to the perturbation innormal load P. Using the Hertzian relation

a = 2

√PRπE∗ (28)

and perturbing about the mean load P0, the followingis obtained

a1 = ∂a∂P

P1 = P1

√R

πP0E∗ = P1a0

2P0(29)

3 OBSERVATION AND ‘TUNING’ OF THESOLUTION

Equations (25) and (29) define the dynamic rela-tionship between the perturbations Q1, P1, and �1



in tangential force, normal force, and rotationalspeed, respectively, and hence define the frequency-dependent receptance of the contact process underthe Winkler spring assumption for tangential contact.Most authors (except those using the Gross-Thebingnumerical technique) obtain an expression for thisreceptance by either assuming arbitrary simplifica-tion of the contact with elastic springs, or by simplyperturbing the appropriate steady-state creep relationabout the mean state, which is equivalent to assum-ing that the steady-state relation applies even undertransient conditions. This assumption is reasonable inthe low frequency limit (ζ → 0) and indeed might beexpected to give reasonable results for moderate val-ues of ζ , since Kalker showed that transient effects inrolling contact become essentially negligible after arolling distance equal to the contact width, but unfor-tunately, short-pitch corrugation does not fall into therange of ‘moderate values of ζ ’. Appropriate values ofthe parameters are λ = 20–80 mm and a0 = 3–5 mm,giving values of ζ in the range 0.4 < ζ < 3! Since thezeroth-order approximation is equivalent (in terms oftangential load) to a dashpot, some authors (e.g. [3])have attempted to improve this zeroth-order approx-imation by adding a contact spring in series with adashpot with parameters obtained from a combina-tion of the Carter and Mindlin solutions. However, thistechnique only fits the tangential load, and nothing issaid about the energy dissipation – the coefficients arenot trivially obtained from those of tangential load, asit will be clearer in the following.

Since the perturbed Carter theory is exact in the limitζ → 0, it is convenient to choose the value for the tan-gential stiffness kq in the present Winkler model suchthat the two theories agree in the limit. The Cartersolution gives the relation

ξ = 1 − �RV

= faR

(1 −

√1 − Q

fP

)(30)

and in the ‘full stick’ limit as f → ∞, this reduces to

Q = √PRπE∗

(1 − �R

V

)(31)

where equation(28) to substitutes for the contact semi-width a. Perturbing this expression with respect to �

gives

Q1 = −�1RV

√P0RπE∗ (32)

This is compared with the second of equation (14),which in the limit ω → 0 tends to

Q1 = −8kqP0R2

πE∗V(33)

from which it is deduced that the two models willpredict the same tangential receptance in the low

frequency limit if

kq = πE∗

8

√πE∗

P0R= πE∗

4a0(34)

Using this result, equations (25) and (26) can bewritten in the form

Q1 = CPP1 + C��1, D1 = DPP1 + D��1 (35)

where

CP(ζ ) = ξ0

2

√πE∗R

P0

[1 − ı

ζ

[1 − exp (−ıζ )

]](36)

C�(ζ ) = − 2V ζ

√πE∗P0R3

{1ζ

[1 − exp (−ıζ )

] − ı

}(37)

DP(ζ ) = V ξ 20

2

√πE∗R

P0

(1 + exp (−ıζ )

)(38)

D�(ζ ) = 2ıξ0

√πE∗P0R3

(1 − exp (−ıζ )

)ζ

(39)

4 COUPLING WITH THE DYNAMICS

To complete the analysis, the rotational equation ofmotion is written for the wheelset, which is simplifiedby omitting stiffness and damping, giving

Iwd�

dt= (Q − Q0) R (40)

where Iw is the inertia of the wheel. It follows that theoscillatory terms are related by the equation

iωIw�1 = Q1R (41)

and substituting for �1 from equation (41) into the firstequation (35) gives

Q1 = CP(ζ )P1 + C�(ζ )Q1RiωI

(42)

Solving for Q1, and using ζ = 2ωa0/V , the tangentialload oscillatory term can be written in the perturba-tion as a function of the oscillatory term in normal loadonly

Q1 = CP(ζ )

1 − (C�(ζ )/ζ )(2Ra0/iIwV )P1 (43)

For dissipation, substituting �1 from equation (41)into the second equation (35) gives

D1 = DP(ζ )P1 + D�(ζ )Q1RiωIw

(44)

which can be expressed in terms of P1 usingequation (43).



Fig. 3 The vertical receptance adapted from Bhaskaret al. [3]. The dotted lines are from a richermodel including the pinned–pinned resonance,not used in the present study

The limit of constant creepage assumed by Grassieand Johnson [4] corresponds to the case where thewheel inertia is very large, or more precisely where thedimensionless parameter

I ≡ IwV 2

P0R3→ ∞

At the other extreme, the assumption of constanttangential load, as in Grassie and Edwards [18] andMeehan et al. [19], corresponds to the case I → 0.

For the normal dynamics (i.e. the relation betweenthe complex amplitude of the perturbation in normalforce P1 and the initial corrugation �), the rail verticalreceptance of Bhaskar et al. [3] is used, which repre-sents a typical BR rail with an equivalent continuoussupport (results from discrete support from Ripke’scalculations for Intercity track having similar param-eters are shown for comparison mid-way betweensleepers (dashed lines) and at a sleeper (dotted lines).

5 RESULTS

The complex dissipation D1, depends on only twoparameters: (i) the inertia of the wheel and (ii) thesteady normal load P0. For the latter, P0 = 50 kN is used

Table 1 The data of Nielsen et al. [[20]]

Munspr R polarI 1/2 M Axle loadTrain (kg) (m)(kg m2) Rˆ2 β (tonnes)

X2 poweredwheelset 1915 0.55 250 290 0.86 18.5

X2 trailerwheelset 1390 0.44 70 134 0.52 12.5

and for the inertia

I = β

2MwR2

where β is a factor ranging in general from 0.5 to 0.8,since the mass of the wheelset is concentrated near thecentre. Nielsen et al. [20] for example report the data inTable 1. Considering that for a single wheel one half ofthe inertia values in the the polar inertia of the entirewheelset is taken, the values seem to range from 35 to125 kg m2, whereas most systems may have somewhatlower values with mass of the wheel around 350 kg andhence inertia in the range 18–30 kg m2.

For completeness and simplicity, consider β = 0.75and two cases:

(a) the mass of the wheel 350 kg, with a normal loadof 50 kN, as a realistic value for a standard systems(where most data on Fig. 1 would likely fall), andcompared with the limit case;

(b) the powered X2 wheelset using 1000 kg of mass asrepresentative of a heavy loaded case with highervalues P0 = 90 kN.

To model the dynamic normal load, the rail recep-tance (from Bhaskar’s model – see Fig. 3), the wheelreceptance, and that of the Hertzian contact springhave to be added. Finally, the correction on the phaseof dissipation resulting from the fact that the high-est dissipation occurs at the rear of the contact, andnot at its centrepoint, is added. Results for steady-state longitudinal creepage are shown, but the valueof the creepage, as well as the friction coefficient andother parameters (like the vertical load) only affect therate of growth of the corrugation, not the preferred(most rapidly growing) wavelength. Hence, the beautyof the qualitative model proposed here is that all theseparameters need not to be fixed.

The dominant wavelength is expected to be thatfor which the predicted wear rate at the troughs isa maximum and this corresponds to the case wherethe dissipation function D1 has the maximum negativereal part. The following figures show 10 equally spacedcontour levels from zero to the maximum negative realpart of dissipation function, as a function of speed andwavelength of corrugation. Coloured lines show con-stant frequency 300, 1000, and 1700 Hz for reference,also equal to the lines in Fig. 1, and the experimentaldata points from Fig. 1 are also included. The bold lines



show the wavelengths corresponding to the maximumnegative real part of D1 as a function of speed V .

Figure 4 shows the result of the standard case A,showing considerable agreement with experiments.Most of the experimental data points fall in a region ofthe possible growth (where the real part of D1 is nega-tive), and most data also seem to concentrate aroundthe bold line of the maxima, which at low speeds doessome zigzaging because the dissipation function isfirst relatively flat, then follows almost a constant fre-quency (around 400 Hz), and then jumps to anotherregime (of high speed and high frequency, crossing the1700 Hz line).

Figure 5, by contrast, shows no significant evidenceof agreement between predictions and experimentaldata.

5.1 Wheel inertia and the two limit assumptions

To compare with the previous theories of Grassie andJohnson [4], Grassie and Edwards [18] and Meehanet al. [19], the limiting cases for constant tangentialload (Fig. 6) and constant creep rate (Fig. 7) are pre-sented by choosing the dimensionless wheel inertia Ito be either very small or very large.

The predictions differ substantially from the exper-imental data in both cases. The results for constanttangential load show some modest qualitative agree-ment, although corrugation is predicted at muchhigher apparent frequency, whereas no substantiveagreement with data is seen for large inertia, as wasalready apparent from Fig. 5.

Fig. 4 The results for case A problem in this paper

Fig. 5 The results for case B – high inertia

Fig. 6 The results for case A under the assumption ofconstant tangential load

6 DISCUSSION

The results in the current paper cannot pretend toanswer all the questions raised by corrugation mecha-nisms and experimental data around, including dif-ferent railways and metro systems. The VancouverSkytrain of which the data on the vertical receptancewas used was continuously supported on a concretebase with a 25 mm continuous rubber pad under



the rail. Periodic clips held the rails in place. Here,pinned–pinned resonance could not be even sug-gested as a mechanism, yet the data in Fig. 1 aresurprisingly ‘similar’ to the data on BR 1911 reportwith probably wooden sleepers, so again pinned–pinned resonance had to be of marginal importancehere. The results suggest that a possible mechanismis longitudinal creepage. That Grassie and Johnson[4] found no mechanism for short-pitch corrugationusing longitudinal creepage was as shown here, dueto the assumption of infinite wheel inertia. This erro-neous conclusion, however, did not stop Frederickfrom making an excellent contribution just 1 yearlater (1986), which included the wheel inertia in bothlateral and longitudinal directions. However, the factthat he found a longitudinal creepage mechanismvery similar to what has been found here has largelyremained obscure, since his regime (at 750 Hz) showeda growth factor 20 times lower than the lateral creepagemechanism, it was only explained for above sleepers,whereas for midsleepers there was again no mecha-nism (notice also that no pinned–pinned resonanceregime was found for longitudinal creepage neither atabove sleepers nor midspan by Frederick). There aremany details of Frederick model which do not per-mit a rapid evaluation, but clearly this factor 20 couldlargely be outweighed by the fact that in the linearregime, growth is proportional to the square of creep-age, and longitudinal creepage can be perhaps be 10times higher than lateral (5 per cent instead of 0.5per cent). Therefore, here there is a factor 100 whichlargely compensates, and suggests again longitudinalcreepage associated with no resonance may be indeed

Fig. 7 The results for for case A under the assumption ofconstant longitudinal creepage

the correct mechanism to interpret the classical data,as first impressions from the comparison seems toindicate.

Later studies by the Berlin group improved manyaspects of the Frederick model, but in one respectthey took a step backwards, in that they did not gen-erally include longitudinal dynamics – whether this isbecause of the original Grassie and Johnson [4] find-ing, or the Frederick result, or neither, is a matter ofspeculation. Hempelmann [9] finds highest growth forlateral creepage at sleepers at pinned–pinned reso-nance, but no growth midspan at this frequency wherehe finds much lesser growth at ‘no-characteristic’frequency.

BR was very active in this area of research and didmany surveys after 1911, which left the impressionthat corrugation on BR was insignificant until theintroduction of continously welded rail and concretesleepers in 1966. The reason for this is still unclear.More investigation is still required to solve this difficultproblem.

7 CONCLUSIONS

A simple two-dimensional model has been developedusing a full-stick Winkler (wire brush) approximationfor the tangential contact problem with a modu-lus chosen to fit the classical Carter solution in thesteady state. Starting from a given small sinusoidalcorrugation profile and realistic values for the verti-cal receptance of the rail, the tangential dynamics ofthe problem were modelled, including the effect offinite wheelset rotational inertia. Corrugation is pre-dicted to grow when the rate of energy dissipation ispositive at the troughs of the original corrugation andthe preferred wavelength is predicted to be that whosedissipation rate at this location is a maximum.

Corrugation from longitudinal creepage is foundto be qualitatively explained in the range of realisticparameter values and the predicted wavelengths arereasonably close to those observed experimentally asa function of train speed. By considering the limits oflarge and small wheelset inertia, it was demonstratedthat the constant creep rate assumption of Grassie andJohnson [4] or Bhaskar et al. [3] could not explain cor-rugation; reasons for more recent successful attemptsassuming constant tangential load have also beengiven.

It can be concluded that it is critical to includethe rotational dynamics of the wheelset in anal-yses of corrugation, and this confirms resultsfound but neglected in previous models (Frederick,Hempelmann, Tassilly and Vincent), which includethe wheel inertia, in that a regime not driven by aresonance of the system (neither of the track or thewheelset) is found possible. Despite that longitudinal



creepage seems to show less growth than lateralcreepage mechanism, the much larger values for lon-gitudinal creepage could largely compensate for thiseffect, and the much better agreement with the exper-imental data of the longitudinal creepage mechanismindicated here than to the ‘pinned–pinned’ resonancelateral mechanism is a strong indication that someconclusions in the literature should be revised. A moreaccurate quantitative model would require a morerealistic contact model, including the effects of partialslip, and this is the subject of an ongoing investigation.

Additionally, the conclusions about pinned–pinnedresonances coming only from numerical models maypartly be erroneous due to difficulties of modellingof the resonances themselves (particularly, damp-ing factors, strongly influencing amplitude response),parametric resonance effects, and perhaps the mosturgent requirement is for more precise experimen-tal data to complete, compare and distinguish thecollection of Fig. 1.

ACKNOWLEDGEMENTS

The authors thank K. L. Johnson and J. Woodhousefrom Cambridge University and S. L. Grassie, for help-ful discussions regarding the present model. Also,Luciano Afferrante for help in the ongoing develop-ment of this model.

REFERENCES

1 Grassie, S. L. and Kalousek, J. Rail corrugations. Charac-teristics, causes and treatments. Proc. Instn Mech. Engrs,Part F: J. Rail and Rapid Transit, 1993, 207, 57–68.

2 Harrison, D. Corrugation of railways. PhD Thesis, Cam-bridge University, UK, 1979.

3 Bhaskar, A., Johnson, K L., Wood, G. D., and Wood-house, J. Wheel-rail dynamics with closely conformalcontact. Part 1: dynamic modelling and stability analysis.Proc. Instn Mech. Engrs, Part F: J. Rail and Rapid Transit,1997, 211, 11–26.

4 Grassie, S. L. and Johnson, K. L. Periodic microslipbetween a rolling wheel and a corrugated rail.Wear, 1985,101, 291–305.

5 Frederick, C. O. A rail corrugation theory. In Proceed-ings of the 2nd International Conference on Contactmechanics of rail–wheel systems, University of RhodeIsland, 1986, pp. 181–211 (University of Waterloo Press,Waterloo, Ontario, Canada).

6 Tassilly, E. and Vincent, N. A linear model for thecorrugation of rails. J. Sound Vibr., 1991, 150(1), 25–45.

7 Tassilly, E. and Vincent, N. Rail corrugations: analyticalmodels and field tests. Wear, 1991, 144(1–2), 163–178.

8 Elkins, J., Grassie, S., and Handal, S. Rail corrugationmitigation in transit, transit cooperative research pro-gram sponsored by the federal transit administration.Res. Results Digest, 1998, 26, 1–33.

9 Hempelmann, K. Short pitch corrugations on railwayrails: a linear model for prediction, 1994 (VDI Fortschritt-Berichte, VDI-Verlag, Dusseldorf).

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11 Gross-Thebing, A. A lineare Modellierung des instation-aren Rollkontaktes von Rad und Schiene.VDI Fortschritt-Berichte, Reihe 12, No. 199, Dusseldort, 1993.

12 Müller, S. A linear wheel-track model to predict instabil-ity and short pitch corrugation. J. Sound Vibr., 1999, 227,899–913.

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18 Grassie, S. L. and Edwards, J. W. Development of corru-gation as a result of varying normal load. In the 7th Inter-national Conference on Contact Mechanics and Wearof Rail/Wheel Systems (CM2006), Brisbane, Australia,24–26 September 2006.

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